L(s) = 1 | + (−0.163 + 1.99i)2-s + 1.56i·3-s + (−3.94 − 0.652i)4-s + 3.43·5-s + (−3.11 − 0.255i)6-s − 2.64i·7-s + (1.94 − 7.75i)8-s + 6.56·9-s + (−0.562 + 6.85i)10-s − 8.48i·11-s + (1.01 − 6.15i)12-s − 18.5·13-s + (5.27 + 0.433i)14-s + 5.36i·15-s + (15.1 + 5.14i)16-s − 8.87·17-s + ⋯ |
L(s) = 1 | + (−0.0818 + 0.996i)2-s + 0.520i·3-s + (−0.986 − 0.163i)4-s + 0.687·5-s + (−0.518 − 0.0425i)6-s − 0.377i·7-s + (0.243 − 0.969i)8-s + 0.729·9-s + (−0.0562 + 0.685i)10-s − 0.771i·11-s + (0.0848 − 0.513i)12-s − 1.42·13-s + (0.376 + 0.0309i)14-s + 0.357i·15-s + (0.946 + 0.321i)16-s − 0.522·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.706467 + 0.599257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.706467 + 0.599257i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.163 - 1.99i)T \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 - 1.56iT - 9T^{2} \) |
| 5 | \( 1 - 3.43T + 25T^{2} \) |
| 11 | \( 1 + 8.48iT - 121T^{2} \) |
| 13 | \( 1 + 18.5T + 169T^{2} \) |
| 17 | \( 1 + 8.87T + 289T^{2} \) |
| 19 | \( 1 - 30.3iT - 361T^{2} \) |
| 23 | \( 1 + 26.5iT - 529T^{2} \) |
| 29 | \( 1 - 18.6T + 841T^{2} \) |
| 31 | \( 1 + 41.2iT - 961T^{2} \) |
| 37 | \( 1 + 3.49T + 1.36e3T^{2} \) |
| 41 | \( 1 - 37.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 51.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 15.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 38.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 72.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 32.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 50.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.48T + 5.32e3T^{2} \) |
| 79 | \( 1 - 39.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 4.28iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 123.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 32.0T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.97613190566524916007627839952, −16.25048873668409978272873322269, −14.90263072485402376771715522757, −13.94839587850568902204141974601, −12.66503136129944813034505453755, −10.34585787492284660047317628124, −9.471623627578727220924373679148, −7.75328382635326175735943251381, −6.12223369836848965921124709915, −4.44786693874506147779299708564,
2.18943478194181923973646081641, 4.90187186608116542117443072336, 7.22273304333977905966444118788, 9.203404833991545109694140593872, 10.18967094449228738159124188043, 11.88053544807426675719344222716, 12.84755382431198112581871324287, 13.84105246619912641141782427265, 15.36693923048420988329348081390, 17.52103065367339427104940124679